Open Up Resources Logo

Lesson 5A New Way to Interpret a over b

Let's investigate what a fraction means when the numerator and denominator are not whole numbers. 

Learning Targets:

  • I understand the meaning of a fraction made up of fractions or decimals, like \frac{2.1}{0.07} or \frac{\frac45}{\frac32} .
  • When I see an equation, I can make up a story that the equation might represent, explain what the variable represents in the story, and solve the equation.

5.1 Recalling Ways of Solving

Solve each equation. Be prepared to explain your reasoning.

  1. 0.07 = 10m  
  1. 10.1 = t + 7.2

5.2 Interpreting \frac{a}{b}

Solve each equation.

  1. 35=7x
  1. 35=11x
  1. 7x=7.7
  1. 0.3x=2.1
  1. \frac25=\frac12 x

Are you ready for more?

Solve the equation. Try to find some shortcuts.

\frac{1}{6} \boldcdot  \frac{3}{20} \boldcdot  \frac{5}{42} \boldcdot  \frac{7}{72} \boldcdot x = \frac{1}{384}

5.3 Storytime Again

Take turns with your partner telling a story that might be represented by each equation. Then, for each equation, choose one story, state what quantity x describes, and solve the equation. If you get stuck, draw a diagram.

  1. 0.7 + x = 12
  1. \frac{1}{4}x = \frac32

Lesson 5 Summary

In the past, you learned that a fraction such as \frac45 can be thought of in a few ways. 

  • \frac45 is a number you can locate on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0.
  • \frac45 is the share that each person would have if 4 wholes were shared equally among 5 people. This means that  \frac45 is the result of dividing 4 by 5.

We can extend this meaning of a fraction as a division to fractions whose numerators and denominators are not whole numbers. For example, we can represent 4.5 pounds of rice divided into portions that each weigh 1.5 pounds as: \frac{4.5}{1.5} = 4.5\div{1.5} = 3 .

Fractions that involve non-whole numbers can also be used when we solve equations. 

Suppose a road under construction is \frac38 finished and the length of the completed part is \frac43 miles. How long will the road be when completed?

We can write the equation \frac38x=\frac43 to represent the situation and solve the equation.

The completed road will be 3\frac59 or about 3.6 miles long.

\begin {align} \frac38x&=\frac43\\[5pt] x&=\frac{\frac43}{\frac38}\\[5pt] x&=\frac43\boldcdot \frac83\\[5pt] x&=\frac{32}{9}=3\frac59\\ \end {align}

Lesson 5 Practice Problems

  1. Select all the expressions that equal \frac{3.15}{0.45} .

    1. (3.15) \boldcdot (0.45)
    2. (3.15) \div (0.45)
    3. (3.15) \boldcdot \frac{1}{0.45}
    4. (3.15) \div \frac{45}{100}
    5. (3.15) \boldcdot \frac{100}{45}
    6. \frac{0.45}{3.15}

  2. Which expressions are solutions to the equation \frac{3}{4}x = 15 ? Select all that apply.

    1. \frac{15}{\frac{3}{4}}
    2. \frac{15}{\frac{4}{3}}
    3. \frac{4}{3} \boldcdot 15
    4. \frac{3}{4} \boldcdot 15
    5. 15 \div \frac{3}{4}

  3. Solve each equation.

    1. 4x = 32
    1. 4=32x
    1. 10x = 26
    1. 26=100x

  4. For each equation, write a story problem represented by the equation. For each equation, state what quantity x represents. If you get stuck, draw a diagram.

    \frac{3}{4} + x = 2

    1.5x = 6

  5. Write as many mathematical expressions or equations as you can about the image. Include a fraction, a decimal number, or a percentage in each.

    A fundraiser thermometer labeled "Fundraiser, our goal, 2 hundred 50 thousand dollars." The numbers 50 thousand through 2 hundred 50 thousand, in increments of 50 thousand dollars, are indicated. There are 4 evenly spaced tick marks between each indicated dollar value. Starting from the bottom, the thermometer is shaded to the first tick mark above 1 hundred thousand dollars.

  6. In a lilac paint mixture, 40% of the mixture is white paint, 20% is blue, and the rest is red. There are 4 cups of blue paint used in a batch of lilac paint. If you get stuck, consider using a tape diagram.

    1. How many cups of white paint are used?
    2. How many cups of red paint are used?
    3. How many cups of lilac paint will this batch yield?

  7. Triangle P has a base of 12 inches and a corresponding height of 8 inches. Triangle Q has a base of 15 inches and a corresponding height of 6.5 inches. Which triangle has a greater area? Show your reasoning.